题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034350-97997.png)
(1)若抛物线过点A、B、C, 求此抛物线的解析式;
(2)求△OAE与△ODC重叠的部分四边形ODFE的面积;
(3)点M是第三象限内抛物线上的一动点,点M在何处时△AMC的面积最大?最大面积是多少?求出此时点
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034350-79987.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-41264.png)
(2)S四边形ODFE=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-66832.png)
(3)当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-13787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-51236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034352-96267.png)
解析
试题分析:(1)由题意易得点A、点B、点C的坐标,利用待定系数法求解即可;
(2)先求出点D及点E的坐标,继而得出直线AE与直线CD的解析式,联立求出点F坐标,根据S四边形ODFE=S△AOE﹣S△ADF,可得出答案.
(3)连接OM,设M点的坐标为(m,n),继而表示出△AMC的面积,利用配方法确定最值,并得出点M的坐标.
试题解析:(1)∵OB=1,OC="3" ,
∴C(0,-3),B(1,0),
∵△OBC绕原点顺时针旋转90°得到△OAE,
∴A(-3,0),
所以抛物线过点A(-3,0),C(0,-3),B(1,0),
设抛物线的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034352-26613.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034352-75988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034352-49482.png)
∴过点A,B,C的抛物线的解析式
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-41264.png)
(2) ∵△OBC绕原点顺时针旋转90°得到△OAE,△OBC沿y轴翻折得到△COD,
∴E(0,-1),D(-1,0),
可求出直线AE的解析式为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034353-30959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034353-72310.png)
∵点F为AE、DC交点,
∴F(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034353-26511.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034353-26511.png)
∴S四边形ODFE=S△AOE-S△ADF=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-66832.png)
(3)连接OM,设M点的坐标为
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034353-54545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034354-63015.png)
∵点M在抛物线上,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034354-11960.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034354-57388.png)
=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034354-46094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034354-25999.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034355-63746.png)
∴当
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-13787.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034351-51236.png)
所以当点M的坐标为(
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034352-96267.png)
核心考点
试题【如图所示,在平面直角坐标系中,Rt△OBC的两条直角边分别落在x轴、y轴上,且OB=1,OC=3,将△OBC绕原点O顺时针旋转90°得到△OAE,将△OBC沿y】;主要考察你对二次函数定义等知识点的理解。[详细]
举一反三
A.y=2(x-1)2-5 | B.y=2(x-1)2+5 |
C.y=2(x+1)2-5 | D.y=2(x+1)2+5 |
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034342-18276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034342-57311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034342-65217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034342-65217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034342-57311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034342-43858.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034343-60282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034337-33874.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034338-73228.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034334-61026.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034335-46931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034335-94270.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034335-59418.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034335-40266.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034326-70327.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034327-28146.jpg)
(1)证明:不论
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034327-43617.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034327-21451.png)
(2)若该函数的图象与
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019034327-24262.png)
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